
thus after restricting to \(\mathcal {U}\) , the distribution of \(x_t\)  conditioned on \(a_t = a\)  is \(\left(\frac{\delta \sigma }{2 A T}\right)\) -smooth with respect to \(\mu _d\) . Thus for each \(a\) , we may apply the regret bound from prop:kpieceregression and, summing over \(a \in \mathcal {A}\)  concludes the proof.

We can now prove prop:conbandits:
[Proof of prop:conbandits]
Note that by Lipschitzness and boundedness, twice the mistake bound is larger than the square loss regret considered in [1]}. Applying [1]} concludes the proof.
